A parallelogram height inequality for Drinfeld modules

TL;DR

This paper establishes a parallelogram height inequality for four interconnected Drinfeld modules, extending known inequalities from abelian varieties to the setting of Drinfeld modules over function fields.

Contribution

It introduces a novel parallelogram inequality for Drinfeld modules, analogous to existing inequalities for abelian varieties, advancing the understanding of their height relations.

Findings

Proves a new height inequality for four Drinfeld modules

Extends parallelogram inequalities from abelian varieties to Drinfeld modules

Provides a framework for analyzing isogeny relations among Drinfeld modules

Abstract

We prove inequalities relating the Taguchi heights, respectively the graded heights, of four Drinfeld modules arranged in a ``parallelogram of isogenies''. This inequality is the analogue for Drinfeld modules of the parallelogram inequality of R\'emond (2022) for abelian varieties over number fields and of Griffon--Le Fourn--Pazuki (2025) for abelian varieties over function fields.